Find the LCM and the GCF of Integers - Examples and Questions with Answers (Grade 5)

Learn how to find the Lowest Common Multiple (LCM) and Greatest Common Factor (GCF) of integers with step-by-step examples and practice questions with answers. Use our interactive calculators below to verify your answers:

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A - Find the Lowest Common Multiple (LCM) of two Numbers

Method 1:
Example 1: Find the LCM of 6 and 4 by listing the multiples.
Find the first few multiples of 6 and 4:
6: 6 , 12 , 18 , 24 , 30 ,...
4: 4 , 8 , 12 , 16 , 20 ,...
The lowest multiple common to both 6 and 4 is 12. Thus, the LCM of 6 and 4 is 12.
Note: This method works best for small numbers.


Method 2:
Find the LCM using prime factorization. Prime factorization uses prime numbers (2, 3, 5, 7, 11, ...) to factor an integer.
Examples of Prime Factorization:
\(4 = 2 \times 2\)
\(14 = 2 \times 7\)
\(16 = 2 \times 2 \times 2 \times 2\)
\(20 = 2 \times 2 \times 5\)

Example 2: Find the LCM of 6 and 4 using prime factorization.
Find the prime factorization of each number:
\(6 = 2 \times 3\)
\(4 = 2 \times 2\)
Multiplying the factored forms:
\(6 \times 4 = (\underline{2} \times 3) \times (\underline{2} \times 2)\)
Since the factor \(\underline{2}\) is counted twice, remove it from one term to get the lowest common multiple:
\(\text{LCM} = 3 \times (2 \times 2) = 12\).


Example 3: Find the LCM of 20 and 24 using prime factorization.
\(20 = 2 \times 2 \times 5\)
\(24 = 2 \times 2 \times 2 \times 3\)
The product in factored form is \((\underline{2} \times \underline{2} \times 5) \times (\underline{2} \times \underline{2} \times 2 \times 3)\).
Removing the duplicate factors \(\underline{2} \times \underline{2}\) from one term gives:
\(\text{LCM} = (5) \times (2 \times 2 \times 2 \times 3) = 120\).


Example 4: Find the LCM of 1240 and 5300 using prime factorization.
\(1240 = \underline{2} \times \underline{2} \times 2 \times \underline{5} \times 31\)
\(5300 = \underline{2} \times \underline{2} \times \underline{5} \times 5 \times 53\)
Removing the duplicate factors \(\underline{2} \times \underline{2} \times \underline{5}\) from one term yields:
\(\text{LCM} = (2 \times 31) \times (\underline{2} \times \underline{2} \times \underline{5} \times 5 \times 53) = 328,600\).


B - Find the Greatest Common Factor (GCF) of two Numbers

Example 5: Find the GCF of 6 and 4 using prime factorization.
\(6 = 2 \times 3\)
\(4 = 2 \times 2\)
The number 2 is the highest common factor shared by both 6 and 4. Thus, \(\text{GCF} = 2\).

Example 6: Find the GCF of 20 and 24 using prime factorization.
\(20 = \underline{2} \times \underline{2} \times 5\)
\(24 = \underline{2} \times \underline{2} \times 2 \times 3\)
The GCF is the product of all common factors: \(\underline{2} \times \underline{2} = 4\).


Example 7: Find the GCF of 1240 and 5300 using prime factorization.
\(1240 = \underline{2} \times \underline{2} \times 2 \times \underline{5} \times 31\)
\(5300 = \underline{2} \times \underline{2} \times \underline{5} \times 5 \times 53\)
The GCF is given by \(\underline{2} \times \underline{2} \times \underline{5} = 20\).


C - Relationship Between The LCM and the GCF of Two Numbers

The table below illustrates that the product of any two whole numbers equals the product of their LCM and GCF.

\((m , n)\) \(\text{LCM}(m,n)\) \(\text{GCF}(m,n)\) \(\text{LCM}(m,n) \times \text{GCF}(m,n)\) \(m \times n\)
\((4 , 6)\) 12 2 \(12 \times 2 = 24\) \(4 \times 6 = 24\)
\((20 , 24)\) 120 4 \(120 \times 4 = 480\) \(20 \times 24 = 480\)

Key Property: For any two integers \(m\) and \(n\):

\(m \times n = \text{LCM}(m,n) \times \text{GCF}(m,n)\)


Questions and Step-by-Step Solutions

Find the LCM and GCF of each pair of numbers and verify that their product equals the product of their LCM and GCF.

  1. Question A: 21 and 14

    View Step-by-Step Solution

    Explanation:

    • Prime Factorization: \(21 = 3 \times 7\) and \(14 = 2 \times 7\).
    • GCF: The common factor is 7. Thus, \(\text{GCF} = 7\).
    • LCM: \(2 \times 3 \times 7 = 42\). Thus, \(\text{LCM} = 42\).
    • Verification: \(\text{GCF} \times \text{LCM} = 7 \times 42 = 294\) and \(21 \times 14 = 294\).

    Answer: \(\text{GCF} = 7\), \(\text{LCM} = 42\)

  2. Question B: 45 and 55

    View Step-by-Step Solution

    Explanation:

    • Prime Factorization: \(45 = 3^2 \times 5\) and \(55 = 5 \times 11\).
    • GCF: The common factor is 5. Thus, \(\text{GCF} = 5\).
    • LCM: \(3^2 \times 5 \times 11 = 9 \times 5 \times 11 = 495\). Thus, \(\text{LCM} = 495\).
    • Verification: \(\text{GCF} \times \text{LCM} = 5 \times 495 = 2475\) and \(45 \times 55 = 2475\).

    Answer: \(\text{GCF} = 5\), \(\text{LCM} = 495\)

  3. Question C: 120 and 248

    View Step-by-Step Solution

    Explanation:

    • Prime Factorization: \(120 = 2^3 \times 3 \times 5\) and \(248 = 2^3 \times 31\).
    • GCF: Common factor is \(2^3 = 8\). Thus, \(\text{GCF} = 8\).
    • LCM: \(2^3 \times 3 \times 5 \times 31 = 3720\). Thus, \(\text{LCM} = 3720\).
    • Verification: \(\text{GCF} \times \text{LCM} = 8 \times 3720 = 29760\) and \(120 \times 248 = 29760\).

    Answer: \(\text{GCF} = 8\), \(\text{LCM} = 3720\)


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